76,174 research outputs found
Extended Symbolic Dynamics in Bistable CML: Existence and Stability of Fronts
We consider a diffusive Coupled Map Lattice (CML) for which the local map is
piece-wise affine and has two stable fixed points. By introducing a
spatio-temporal coding, we prove the one-to-one correspondence between the set
of global orbits and the set of admissible codes. This relationship is applied
to the study of the (uniform) fronts' dynamics. It is shown that, for any given
velocity in , there is a parameter set for which the fronts with that
velocity exist and their shape is unique. The dependence of the velocity of the
fronts on the local map's discontinuity is proved to be a Devil's staircase.
Moreover, the linear stability of the global orbits which do not reach the
discontinuity follows directly from our simple map. For the fronts, this
statement is improved and as a consequence, the velocity of all the propagating
interfaces is computed for any parameter. The fronts are shown to be also
nonlinearly stable under some restrictions on the parameters. Actually, these
restrictions follow from the co-existence of uniform fronts and non-uniformly
travelling fronts for strong coupling. Finally, these results are extended to
some local maps.Comment: 27 pages, Latex, 2 figure
Stability criteria for planar linear systems with state reset
In this work we perform a stability analysis for a class of switched linear systems, modeled as hybrid automata. We deal with a switched linear planar system, modeled by a hybrid automaton with one discrete state. We assume the guard on the transition is a line in the state space and the reset map is a linear projection onto the x-axis. We define necessary and sufficient conditions for stability of the switched linear system with fixed and arbitrary dynamics in the location. \u
Fifty Years of Stiffness
The notion of stiffness, which originated in several applications of a
different nature, has dominated the activities related to the numerical
treatment of differential problems for the last fifty years. Contrary to what
usually happens in Mathematics, its definition has been, for a long time, not
formally precise (actually, there are too many of them). Again, the needs of
applications, especially those arising in the construction of robust and
general purpose codes, require nowadays a formally precise definition. In this
paper, we review the evolution of such a notion and we also provide a precise
definition which encompasses all the previous ones.Comment: 24 pages, 11 figure
High order operator splitting methods based on an integral deferred correction framework
Integral deferred correction (IDC) methods have been shown to be an efficient
way to achieve arbitrary high order accuracy and possess good stability
properties. In this paper, we construct high order operator splitting schemes
using the IDC procedure to solve initial value problems (IVPs). We present
analysis to show that the IDC methods can correct for both the splitting and
numerical errors, lifting the order of accuracy by with each correction,
where is the order of accuracy of the method used to solve the correction
equation. We further apply this framework to solve partial differential
equations (PDEs). Numerical examples in two dimensions of linear and nonlinear
initial-boundary value problems are presented to demonstrate the performance of
the proposed IDC approach.Comment: 33 pages, 22 figure
Robust constrained model predictive control based on parameter-dependent Lyapunov functions
The problem of robust constrained model predictive control (MPC) of systems with polytopic uncertainties is considered in this paper. New sufficient conditions for the existence of parameter-dependent Lyapunov functions are proposed in terms of linear matrix inequalities (LMIs), which will reduce the conservativeness resulting from using a single Lyapunov function. At each sampling instant, the corresponding parameter-dependent Lyapunov function is an upper bound for a worst-case objective function, which can be minimized using the LMI convex optimization approach. Based on the solution of optimization at each sampling instant, the corresponding state feedback controller is designed, which can guarantee that the resulting closed-loop system is robustly asymptotically stable. In addition, the feedback controller will meet the specifications for systems with input or output constraints, for all admissible time-varying parameter uncertainties. Numerical examples are presented to demonstrate the effectiveness of the proposed techniques
Stochastic ordinary differential equations in applied and computational mathematics
Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation
Phase limitations of Zames-Falb multipliers
Phase limitations of both continuous-time and discrete-time Zames-Falb
multipliers and their relation with the Kalman conjecture are analysed. A phase
limitation for continuous-time multipliers given by Megretski is generalised
and its applicability is clarified; its relation to the Kalman conjecture is
illustrated with a classical example from the literature. It is demonstrated
that there exist fourth-order plants where the existence of a suitable
Zames-Falb multiplier can be discarded and for which simulations show unstable
behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is
developed. Its application is demonstrated with a second-order counterexample
to the Kalman conjecture. Finally, the discrete-time limitation is used to show
that there can be no direct counterpart of the off-axis circle criterion in the
discrete-time domain
Robust MPC of constrained nonlinear systems based on interval arithmetic
A robust MPC for constrained discrete-time nonlinear systems with additive
uncertainties is presented. The proposed controller is based on the concept of reachable sets, that
is, the sets that contain the predicted evolution of the uncertain system for all possible uncertainties.
If processes are nonlinear these sets are very difficult to compute. A conservative approximation
based on interval arithmetic is proposed for the online computation of these sets. This technique
provides good results with a computational effort only slightly greater than the one corresponding to
the nominal prediction. These sets are incorporated into the MPC formulation to achieve robust
stability. By choosing a robust positively invariant set as a terminal constraint, a robustly stabilising
controller is obtained. Stability is guaranteed in the case of suboptimality of the computed solution.
The proposed controller is applied to a continuous stirred tank reactor with an exothermic reaction.Ministerio de Ciencia y Tecnología DPI-2001-2380-03- 01Ministerio de Ciencia y Tecnología DPI-2002-4375-C02-0
The Time Invariance Principle, Ecological (Non)Chaos, and A Fundamental Pitfall of Discrete Modeling
This paper is to show that most discrete models used for population dynamics
in ecology are inherently pathological that their predications cannot be
independently verified by experiments because they violate a fundamental
principle of physics. The result is used to tackle an on-going controversy
regarding ecological chaos. Another implication of the result is that all
continuous dynamical systems must be modeled by differential equations. As a
result it suggests that researches based on discrete modeling must be closely
scrutinized and the teaching of calculus and differential equations must be
emphasized for students of biology
Stabilizing switching signals: a transition from point-wise to asymptotic conditions
Characterization of classes of switching signals that ensure stability of
switched systems occupies a significant portion of the switched systems
literature. This article collects a multitude of stabilizing switching signals
under an umbrella framework. We achieve this in two steps: Firstly, given a
family of systems, possibly containing unstable dynamics, we propose a new and
general class of stabilizing switching signals. Secondly, we demonstrate that
prior results based on both point-wise and asymptotic characterizations follow
our result. This is the first attempt in the switched systems literature where
these switching signals are unified under one banner.Comment: 7 page
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